Class 12 Applications Of Derivatives · Long Answer3 Marksmonotonicity

Class 12 Maths · Applications Of Derivatives · Long Answer #57

Prove f(x)=2x+arccot x+ln(√(1+x²)−x) is strictly increasing on R. Use arccot with range (0,π).

✅ Step-by-Step Solution

Final Answer: Strictly increasing on R.

√(1+x²)>|x| makes the logarithm argument positive. Its derivative is −1/√(1+x²).
Thus f′=2−1/(1+x²)−1/√(1+x²)=[1+2x²−√(1+x²)]/(1+x²).
Both sides of 1+2x²≥√(1+x²) are positive; their squared difference is 4x⁴+3x²≥0, with equality only at 0.
Therefore f′≥0 and is positive except at 0, proving strict increase on R.

[0.5 marks] √(1+x²)>|x| makes the logarithm argument positive. Its derivative is −1/√(1+x²).
[1 mark] Thus f′=2−1/(1+x²)−1/√(1+x²)=[1+2x²−√(1+x²)]/(1+x²).
[1 mark] Both sides of 1+2x²≥√(1+x²) are positive; their squared difference is 4x⁴+3x²≥0, with equality only at 0.
[0.5 marks] Therefore f′≥0 and is positive except at 0, proving strict increase on R.

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